On the classification of normal G-varieties with spherical orbits

TL;DR

This paper provides a comprehensive geometric and combinatorial framework for classifying normal G-varieties with spherical orbits, extending classical spherical variety theory and recent T-variety developments.

Contribution

It develops a unified approach based on Luna-Vust theory to describe all normal G-varieties with spherical orbits, generalizing existing theories.

Findings

Unified description of G-varieties with spherical orbits

Extension of classical spherical variety theory

Connection to T-varieties and Luna-Vust framework

Abstract

In this article, we investigate the geometry of reductive group actions on algebraic varieties. Given a connected reductive group $G$, we elaborate on a geometric and combinatorial approach based on Luna-Vust theory to describe every normal $G$-variety with spherical orbits. This description encompasses the classical case of spherical varieties and the theory of $\mathbb{T}$-varieties recently introduced by Altmann, Hausen, and S\"uss.